Properties

Degree $2$
Conductor $1344$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 18·5-s − 7·7-s + 9·9-s − 44·11-s − 58·13-s − 54·15-s − 130·17-s − 92·19-s + 21·21-s + 84·23-s + 199·25-s − 27·27-s + 250·29-s − 72·31-s + 132·33-s − 126·35-s + 354·37-s + 174·39-s + 334·41-s + 416·43-s + 162·45-s − 464·47-s + 49·49-s + 390·51-s + 450·53-s − 792·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.60·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.23·13-s − 0.929·15-s − 1.85·17-s − 1.11·19-s + 0.218·21-s + 0.761·23-s + 1.59·25-s − 0.192·27-s + 1.60·29-s − 0.417·31-s + 0.696·33-s − 0.608·35-s + 1.57·37-s + 0.714·39-s + 1.27·41-s + 1.47·43-s + 0.536·45-s − 1.44·47-s + 1/7·49-s + 1.07·51-s + 1.16·53-s − 1.94·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{1344} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.594504844\)
\(L(\frac12)\) \(\approx\) \(1.594504844\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + p T \)
7 \( 1 + p T \)
good5 \( 1 - 18 T + p^{3} T^{2} \)
11 \( 1 + 4 p T + p^{3} T^{2} \)
13 \( 1 + 58 T + p^{3} T^{2} \)
17 \( 1 + 130 T + p^{3} T^{2} \)
19 \( 1 + 92 T + p^{3} T^{2} \)
23 \( 1 - 84 T + p^{3} T^{2} \)
29 \( 1 - 250 T + p^{3} T^{2} \)
31 \( 1 + 72 T + p^{3} T^{2} \)
37 \( 1 - 354 T + p^{3} T^{2} \)
41 \( 1 - 334 T + p^{3} T^{2} \)
43 \( 1 - 416 T + p^{3} T^{2} \)
47 \( 1 + 464 T + p^{3} T^{2} \)
53 \( 1 - 450 T + p^{3} T^{2} \)
59 \( 1 - 516 T + p^{3} T^{2} \)
61 \( 1 + 58 T + p^{3} T^{2} \)
67 \( 1 - 656 T + p^{3} T^{2} \)
71 \( 1 + 940 T + p^{3} T^{2} \)
73 \( 1 - 178 T + p^{3} T^{2} \)
79 \( 1 - 1072 T + p^{3} T^{2} \)
83 \( 1 + 660 T + p^{3} T^{2} \)
89 \( 1 - 1254 T + p^{3} T^{2} \)
97 \( 1 - 210 T + p^{3} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.389286676420585866449983805115, −8.641018581485913260552765361852, −7.41299269133279712686684744999, −6.54623337872490097770516560508, −6.02421558732699749923671979836, −5.05316028380775721225845610444, −4.48729041780794812450101521517, −2.50768422391814310217388281376, −2.31954006524157926604557519878, −0.61828444169259880077539345329, 0.61828444169259880077539345329, 2.31954006524157926604557519878, 2.50768422391814310217388281376, 4.48729041780794812450101521517, 5.05316028380775721225845610444, 6.02421558732699749923671979836, 6.54623337872490097770516560508, 7.41299269133279712686684744999, 8.641018581485913260552765361852, 9.389286676420585866449983805115

Graph of the $Z$-function along the critical line